3.11.55 \(\int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac {8 x}{35 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{35 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{35 (1-x)^{5/2} \sqrt {x+1}}+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}} \]

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Rubi [A]  time = 0.01, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {45, 39} \begin {gather*} \frac {8 x}{35 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{35 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{35 (1-x)^{5/2} \sqrt {x+1}}+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

1/(7*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(3/2)*Sqrt[1 + x]) + (8*x)/
(35*Sqrt[1 - x]*Sqrt[1 + x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{7} \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {12}{35} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8}{35} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{35 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 0.49 \begin {gather*} \frac {8 x^4-24 x^3+20 x^2+4 x-13}{35 (x-1)^3 \sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

(-13 + 4*x + 20*x^2 - 24*x^3 + 8*x^4)/(35*(-1 + x)^3*Sqrt[1 - x^2])

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IntegrateAlgebraic [A]  time = 0.08, size = 76, normalized size = 0.93 \begin {gather*} \frac {(x+1)^{7/2} \left (-\frac {35 (1-x)^4}{(x+1)^4}+\frac {140 (1-x)^3}{(x+1)^3}+\frac {70 (1-x)^2}{(x+1)^2}+\frac {28 (1-x)}{x+1}+5\right )}{560 (1-x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

((1 + x)^(7/2)*(5 - (35*(1 - x)^4)/(1 + x)^4 + (140*(1 - x)^3)/(1 + x)^3 + (70*(1 - x)^2)/(1 + x)^2 + (28*(1 -
x))/(1 + x)))/(560*(1 - x)^(7/2))

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fricas [A]  time = 1.27, size = 86, normalized size = 1.05 \begin {gather*} \frac {13 \, x^{5} - 39 \, x^{4} + 26 \, x^{3} + 26 \, x^{2} - {\left (8 \, x^{4} - 24 \, x^{3} + 20 \, x^{2} + 4 \, x - 13\right )} \sqrt {x + 1} \sqrt {-x + 1} - 39 \, x + 13}{35 \, {\left (x^{5} - 3 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} - 3 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/35*(13*x^5 - 39*x^4 + 26*x^3 + 26*x^2 - (8*x^4 - 24*x^3 + 20*x^2 + 4*x - 13)*sqrt(x + 1)*sqrt(-x + 1) - 39*x
+ 13)/(x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*x + 1)

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giac [A]  time = 0.70, size = 79, normalized size = 0.96 \begin {gather*} \frac {\sqrt {2} - \sqrt {-x + 1}}{32 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{32 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} - \frac {{\left ({\left ({\left (93 \, x - 523\right )} {\left (x + 1\right )} + 1400\right )} {\left (x + 1\right )} - 1120\right )} \sqrt {x + 1} \sqrt {-x + 1}}{560 \, {\left (x - 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

1/32*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/32*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 1/560*(((93*x - 523)*(
x + 1) + 1400)*(x + 1) - 1120)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4

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maple [A]  time = 0.00, size = 35, normalized size = 0.43 \begin {gather*} -\frac {8 x^{4}-24 x^{3}+20 x^{2}+4 x -13}{35 \sqrt {x +1}\, \left (-x +1\right )^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+1)^(9/2)/(x+1)^(3/2),x)

[Out]

-1/35*(8*x^4-24*x^3+20*x^2+4*x-13)/(x+1)^(1/2)/(-x+1)^(7/2)

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maxima [B]  time = 1.38, size = 134, normalized size = 1.63 \begin {gather*} \frac {8 \, x}{35 \, \sqrt {-x^{2} + 1}} - \frac {1}{7 \, {\left (\sqrt {-x^{2} + 1} x^{3} - 3 \, \sqrt {-x^{2} + 1} x^{2} + 3 \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} + \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

8/35*x/sqrt(-x^2 + 1) - 1/7/(sqrt(-x^2 + 1)*x^3 - 3*sqrt(-x^2 + 1)*x^2 + 3*sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))
+ 4/35/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x + sqrt(-x^2 + 1)) - 4/35/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))

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mupad [B]  time = 0.36, size = 68, normalized size = 0.83 \begin {gather*} -\frac {4\,x\,\sqrt {1-x}-13\,\sqrt {1-x}+20\,x^2\,\sqrt {1-x}-24\,x^3\,\sqrt {1-x}+8\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(9/2)*(x + 1)^(3/2)),x)

[Out]

-(4*x*(1 - x)^(1/2) - 13*(1 - x)^(1/2) + 20*x^2*(1 - x)^(1/2) - 24*x^3*(1 - x)^(1/2) + 8*x^4*(1 - x)^(1/2))/(3
5*(x - 1)^4*(x + 1)^(1/2))

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sympy [B]  time = 44.94, size = 423, normalized size = 5.16 \begin {gather*} \begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 \sqrt {-1 + \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 i \sqrt {1 - \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(9/2)/(1+x)**(3/2),x)

[Out]

Piecewise((-8*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560
) + 56*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 140
*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 140*sqrt(
-1 + 2/(x + 1))*(x + 1)/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 35*sqrt(-1 + 2/(x
+ 1))/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560), 2/Abs(x + 1) > 1), (-8*I*sqrt(1 - 2/(
x + 1))*(x + 1)**4/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 56*I*sqrt(1 - 2/(x + 1)
)*(x + 1)**3/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 140*I*sqrt(1 - 2/(x + 1))*(x
+ 1)**2/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 140*I*sqrt(1 - 2/(x + 1))*(x + 1)/
(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 35*I*sqrt(1 - 2/(x + 1))/(-1120*x + 35*(x
+ 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560), True))

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